Constructor for Right Matrix Fraction Descriptions (RMFDs)

A Right Matrix Fraction Description (RMFD) of a rational matrix, \(k(z)\) say, is pair \((c(z),d(z))\) of polynomial matrices, such that \(k(z) = d(z) c^{-1}(z)\). The polynomial matrix \(c(z)\) must be square and invertible.

Usage

rmfd(c = NULL, d = NULL)

Arguments

c, d

polm objects, or objects which may be coerced to a polm object, via x = polm(x). Either of the two arguments may be omitted.

Value

An object of class rmfd.

Details

Suppose that \(k(z) = d(z) c^{-1}(z)\) is an \((m,n)\)-dimensional matrix and that \(c(z)\) and \(d(z)\) have degrees \(p\) and \(q\) respectively. The corresponding rmfd object stores the coefficients of the polynomials \(c(z), d(z)\) in an \((n(p+1)+m(q+1), n)\) dimensional (real or complex valued) matrix. The rmfd object also stores the attribute order = c(m,n,p,q) and a class attribute c("lmfd", "ratm").

For a valid RMFD we require \(m>0\) and \(p\geq 0\).

See also

Useful methods and functions for the rmfd class are:

• test_rmfd generates random rational matrices in RMFD form.

• checks: is.rmfd, is.miniphase, is.stable and is.coprime.

• generic S3 methods: dim, str and print.

• arithmetics: Ops.ratm.

• matrix operations: bind.

• extract the factors \(c(z)\) and \(d(z)\) with $.rmfd.

• zeroes, pseries, zvalues, ...

Examples

### (1 x 1) rational matrix k(z) =  (3+2z+z^2) (1+z+z^2)^(-1)
rmfd(c(1,1,1), c(3,2,1)) %>% print(format = 'c')
#> ( 1 x 1 ) right matrix fraction description d(z) c^(-1)(z) with degrees deg(c(z)) = p = 2, deg(d(z)) = q = 2
#> left factor d(z):
#>               [,1]
#> [1,]  3 + 2z + z^2 
#> right factor c(z):
#>              [,1]
#> [1,]  1 + z + z^2 

### (1 x 1) rational matrix k(z) = (3+2z+z^2)
rmfd(c = c(3,2,1)) %>% print(format = 'c')
#> ( 1 x 1 ) right matrix fraction description d(z) c^(-1)(z) with degrees deg(c(z)) = p = 2, deg(d(z)) = q = 0
#> left factor d(z):
#>       [,1]
#> [1,]     1 
#> right factor c(z):
#>               [,1]
#> [1,]  3 + 2z + z^2 

### (1 x 1) rational matrix k(z) = (1+z+z^2)^(-1)
rmfd(c(1,1,1)) %>% print(format = 'c')
#> ( 1 x 1 ) right matrix fraction description d(z) c^(-1)(z) with degrees deg(c(z)) = p = 2, deg(d(z)) = q = 0
#> left factor d(z):
#>       [,1]
#> [1,]     1 
#> right factor c(z):
#>              [,1]
#> [1,]  1 + z + z^2 

### (3 x 2) rational matrix with degrees p=1, q=1
x = rmfd(array(rnorm(2*2*2), dim = c(2,2,2)), 
         array(rnorm(3*2*2), dim = c(3,2,2)))
is.rmfd(x)
#> [1] TRUE
dim(x)
#> m n p q 
#> 3 2 1 1 
str(x)
#> ( 3 x 2 ) right matrix fraction description with degrees (deg(c(z)) = p = 1, deg(d(z)) = q = 1)
print(x, digits = 2)
#> ( 3 x 2 ) right matrix fraction description d(z) c^(-1)(z) with degrees deg(c(z)) = p = 1, deg(d(z)) = q = 1
#> left factor d(z):
#>      z^0 [,1]  [,2] z^1 [,1]  [,2]
#> [1,]    -0.80  1.90    -1.65 -0.95
#> [2,]    -1.03  0.26    -1.20 -0.03
#> [3,]    -0.35  0.64     0.77  0.35
#> right factor c(z):
#>      z^0 [,1]  [,2] z^1 [,1]  [,2]
#> [1,]    -1.25 -0.53     0.16 -0.60
#> [2,]     0.16  0.55    -0.68  1.64

xr = test_rmfd(dim = c(3,2), degree = c(2,2))

if (FALSE) {
### the following calls to rmfd() throw an error 
rmfd() # no arguments!
rmfd(c = test_polm(dim = c(2,3), degree = 1))  # c(z) must be square 
rmfd(c = test_polm(dim = c(2,2), degree = -1)) # c(z) must have degree >= 0
}